A057557 -id:A057557 - cp's OEIS frontend

A186003 Distance array associated with ordering A057557 of N X N X N, by antidiagonals (distances to yz plane).

1, 2, 4, 3, 8, 10, 5, 9, 18, 20, 6, 15, 19, 33, 35, 7, 16, 30, 34, 54, 56, 11, 17, 31, 51, 55, 82, 84, 12, 26, 32, 52, 79, 83, 118, 120, 13, 27, 47, 53, 80, 115, 119, 163, 165, 14, 28, 48, 75, 81, 116, 160, 164, 218, 220, 21, 29, 49, 76, 111, 117, 161, 215, 219, 284, 286, 22, 42, 50, 77, 112, 156, 162, 216, 281, 285, 362, 364

Offset: 1

Clark Kimberling, Feb 10 2011

Let n=n(i,j,k) be the position of (i,j,k) in the lexicographic ordering A057557 of N X N X N, where N={1,2,3,...}. Row h of A186003 lists those n for which i=n, the distance from (i,j,k) to the yz-plane. Every positive integer occurs exactly once in the array, so that as a sequence, A186003 is a permutation of the positive integers.

			Northwest corner:
   1,  2,  3,  5,  6,  7,  11
   4,  8,  9, 15, 16, 17,  26
  10, 18, 19, 30, 31, 32,  47
  20, 33, 34, 51, 52, 53,  75
  35, 54, 55, 79, 80, 81, 111
T(2,1)=4, the position of (2,1,1) in the ordering
(1,1,1) < (1,1,2) < (1,2,1) < (2,1,1) < (1,1,3) < (1,2,2) < (1,3,1) < ...

G. C. Greubel, Table of n, a(n) for the first 50 rows, flattened

Cf. A057557, A186004, A186005.

lexicographicLattice[{dim_,maxHeight_}]:=Flatten[Array[Sort@Flatten[(Permutations[#1]&)/@IntegerPartitions[#1+dim-1,{dim}],1]&,maxHeight],1];
lexicographicLatticeHeightArray[{dim_,maxHeight_,axis_}]:=Array[Flatten@Position[Map[#[[axis]]&,lexicographicLattice[{dim,maxHeight}]],#]&,maxHeight];
llha=lexicographicLatticeHeightArray[{3,12,1}];
ordering=lexicographicLattice[{2,Length[llha]}];
llha[[#1,#2]]&@@#1&/@ordering
(* Peter J. C. Moses, Feb 15 2011 *)

A186004 Distance array associated with ordering A057557 of N X N X N, by antidiagonals (distances to xz plane).

Original entry on oeis.org

1, 2, 3, 4, 6, 7, 5, 9, 13, 14, 8, 12, 17, 24, 25, 10, 16, 23, 29, 40, 41, 11, 19, 28, 39, 46, 62, 63, 15, 22, 32, 45, 61, 69, 91, 92, 18, 27, 38, 50, 68, 90, 99, 128, 129, 20, 31, 44, 60, 74, 98, 127, 137, 174, 175, 21, 34, 49, 67, 89, 105, 136, 173, 184, 230, 231, 26, 37, 53, 73, 97, 126, 144, 183, 229, 241, 297, 298

Offset: 1

Clark Kimberling, Feb 10 2011

Let n=n(i,j,k) be the position of (i,j,k) in the lexicographic ordering A057557 of N X N X N, where N={1,2,3,...}. Row h of A186004 lists those n for which j=n, the distance from (i,j,k) to the xz-plane. Every positive integer occurs exactly once in the array, so that as a sequence, A186004 is a permutation of the positive integers.

			Northwest corner:
   1,  2,  4,  5,  8, 10
   3,  6,  9, 12, 16, 19
   7, 13, 17, 23, 28, 32
  14, 24, 29, 39, 45, 50
  25, 40, 46, 61, 68, 74
T(2,2)=6, the position of (1,2,2) in the ordering
(1,1,1) < (1,1,2) < (1,2,1) < (2,1,1) < (1,1,3) < (1,2,2) < (1,3,1) < ...

G. C. Greubel, Table of n, a(n) for the first 50 rows, flattened

Cf. A057557, A186003, A186005.

lexicographicLattice[{dim_,maxHeight_}]:=Flatten[Array[Sort@Flatten[(Permutations[#1]&)/@IntegerPartitions[#1+dim-1,{dim}],1]&,maxHeight],1];
lexicographicLatticeHeightArray[{dim_,maxHeight_,axis_}]:=Array[Flatten@Position[Map[#[[axis]]&,lexicographicLattice[{dim,maxHeight}]],#]&,maxHeight];
llha=lexicographicLatticeHeightArray[{3,12,2}];
ordering=lexicographicLattice[{2,Length[llha]}];
llha[[#1,#2]]&@@#1&/@ordering
(* Peter J. C. Moses, Feb 15 2011 *)

A186005 Distance array associated with ordering A057557 of N X N X N by antidiagonals (distances to xy plane).

Original entry on oeis.org

1, 3, 2, 4, 6, 5, 7, 8, 12, 11, 9, 13, 15, 22, 21, 10, 16, 23, 26, 37, 36, 14, 18, 27, 38, 42, 58, 57, 17, 24, 30, 43, 59, 64, 86, 85, 19, 28, 39, 47, 65, 87, 93, 122, 121, 20, 31, 44, 60, 70, 94, 123, 130, 167, 166, 25, 33, 48, 66, 88, 100, 131, 168, 176, 222, 221, 29, 40, 51, 71, 95, 124, 138, 177, 223, 232, 288, 287

Offset: 1

Clark Kimberling, Feb 10 2011

Let n=n(i,j,k) be the position of (i,j,k) in the lexicographic ordering A057557 of N X N X N, where N={1,2,3,...}. Row h of A186005 lists those n for which k=n, the distance from (i,j,k) to the xy-plane. Every positive integer occurs exactly once in the array, so that as a sequence, A186005 is a permutation of the positive integers.

			T(2,2)=6, the position of (1,2,2) in the ordering
(1,1,1) < (1,1,2) < (1,2,1) < (2,1,1) < (1,1,3) < (1,2,2) < (1,3,1) < ...
Northwest corner:
   1,  3,  4,  7,  9, 10
   2,  6,  8, 13, 16, 18
   5, 12, 15, 23, 27, 30
  11, 22, 26, 38, 43, 47
  21, 37, 42, 59, 65, 70

G. C. Greubel, Table of n, a(n) for the first 50 rows, flattened

Cf. A057557, A186003, A186004.

lexicographicLattice[{dim_,maxHeight_}]:=Flatten[Array[Sort@Flatten[(Permutations[#1]&)/@IntegerPartitions[#1+dim-1,{dim}],1]&,maxHeight],1];
lexicographicLatticeHeightArray[{dim_,maxHeight_,axis_}]:=Array[Flatten@Position[Map[#[[axis]]&,lexicographicLattice[{dim,maxHeight}]],#]&,maxHeight];
llha=lexicographicLatticeHeightArray[{3,12,3}];
ordering=lexicographicLattice[{2,Length[llha]}];
llha[[#1,#2]]&@@#1&/@ordering
(* Peter J. C. Moses, Feb 15 2011 *)

A057555 Lexicographic ordering of N x N, where N = {1,2,3...}.

Original entry on oeis.org

1, 1, 1, 2, 2, 1, 1, 3, 2, 2, 3, 1, 1, 4, 2, 3, 3, 2, 4, 1, 1, 5, 2, 4, 3, 3, 4, 2, 5, 1, 1, 6, 2, 5, 3, 4, 4, 3, 5, 2, 6, 1, 1, 7, 2, 6, 3, 5, 4, 4, 5, 3, 6, 2, 7, 1, 1, 8, 2, 7, 3, 6, 4, 5, 5, 4, 6, 3, 7, 2, 8, 1, 1, 9, 2, 8, 3, 7, 4, 6, 5, 5, 6, 4, 7, 3, 8, 2, 9, 1, 1, 10, 2, 9, 3, 8, 4, 7, 5, 6, 6, 5, 7, 4, 8, 3, 9, 2, 10, 1, 1, 11, 2, 10, 3, 9, 4, 8, 5, 7, 6, 6, 7, 5, 8, 4, 9, 3, 10, 2, 11, 1, 1, 12, 2, 11, 3, 10, 4, 9, 5, 8, 6, 7, 7, 6, 8, 5, 9, 4, 10, 3, 11, 2, 12, 1

Offset: 1

Clark Kimberling, Sep 07 2000

nonn

			Flatten the ordered lattice points (1,1) < (1,2) < (2,1) < (1,3) < (2,2) < ... as 1,1, 1,2, 2,1, 1,3, 2,2, ...

Cf. A057554, A057557, A057559.

Cf. A181118.

lexicographicLattice[{dim_,maxHeight_}]:= Flatten[Array[Sort@Flatten[(Permutations[#1]&)/@IntegerPartitions[#1+dim-1,{dim}],1]&,maxHeight],1]; Flatten@lexicographicLattice[{2,12}] (* Peter J. C. Moses, Feb 10 2011 *)
u[x_] := Floor[3/2 + Sqrt[2*x]]; v[x_] := Floor[1/2 + Sqrt[2*x]]; n[x_] := x - v[x]*(v[x] - 1)/2; k[x_] := 1 - x + u[x]*(u[x] - 1)/2; Flatten[Table[{n[m], k[m]}, {m, 45}]] (* L. Edson Jeffery, Jun 20 2015 *)

a(n)= if(n<1, 0, 1+(-1)^(n%2) * (binomial((n+1)%2+(sqrtint(4*n)+1)\2, 2)-n\2)) /* Michael Somos, Mar 06 2004 */

a(2n) = A004736(n), a(2n+1) = A002260(n). - Michael Somos, Mar 06 2004

Let p(i,j) be the position of (i,j) in the ordering. Then p(i,j) = ((i+j)^2-i-3j+2)/2. Inversely, the pair (i,j) in a given position p is given by i=p-q(q-1)/2 and j=q+1-i, where q=floor((1+sqrt(8k-7))/2).

Extended by Clark Kimberling, Feb 10 2011

A057556 Lexicographic ordering of M x M x M, where M={0,1,2,...}.

Original entry on oeis.org

0, 0, 0, 0, 0, 1, 0, 1, 0, 1, 0, 0, 0, 0, 2, 0, 1, 1, 0, 2, 0, 1, 0, 1, 1, 1, 0, 2, 0, 0, 0, 0, 3, 0, 1, 2, 0, 2, 1, 0, 3, 0, 1, 0, 2, 1, 1, 1, 1, 2, 0, 2, 0, 1, 2, 1, 0, 3, 0, 0, 0, 0, 4, 0, 1, 3, 0, 2, 2, 0, 3, 1, 0, 4, 0, 1, 0, 3, 1, 1, 2, 1, 2, 1, 1, 3, 0, 2, 0, 2, 2, 1, 1, 2, 2, 0, 3, 0, 1, 3, 1, 0, 4, 0, 0, 0, 0, 5, 0, 1, 4, 0, 2, 3, 0, 3, 2, 0, 4, 1, 0, 5, 0, 1, 0, 4, 1, 1, 3, 1, 2, 2, 1, 3, 1, 1, 4, 0, 2, 0, 3, 2, 1, 2, 2, 2, 1, 2, 3, 0, 3, 0, 2, 3, 1, 1, 3, 2, 0, 4, 0, 1, 4, 1, 0, 5, 0, 0

Offset: 1

Clark Kimberling, Sep 07 2000

See A057557 for N x N x N, where N={1,2,3,...}.

The triples are sorted first according to their sum, then lexicographically. - Pontus von Brömssen, Aug 16 2023

			Flatten the list of ordered lattice points, (0,0,0) < (0,0,1) < (0,1,0) < ... to 0,0,0, 0,0,1, 0,1,0, ...
As a three-column array:
  0 0 0
  0 0 1
  0 1 0
  1 0 0
  0 0 2
  0 1 1
  0 2 0
  1 0 1
  1 1 0
  2 0 0
  0 0 3
  0 1 2
  0 2 1
  0 3 0
  1 0 2
  1 1 1
  1 2 0
  2 0 1
  2 1 0
  3 0 0
  ...

Alois P. Heinz, Table of n, a(n) for n = 1..10962

Cf. A057554, A057557.

Cf. A144625 (each triple reversed).

lexicographicLattice[{dim_,maxHeight_}]:= Flatten[Array[Sort@Flatten[(Permutations[#1]&)/@IntegerPartitions[#1+dim-1,{dim}],1]&,maxHeight],1]; Flatten@lexicographicLattice[{3,6}]-1
(* Peter J. C. Moses, Feb 10 2011 *)

Extended by Clark Kimberling, Feb 10 2011

A057559 Lexicographic ordering of NxNxNxN, where N={1,2,3,...}.

Original entry on oeis.org

1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 2, 1, 1, 2, 1, 1, 2, 1, 1, 1, 1, 1, 1, 3, 1, 1, 2, 2, 1, 1, 3, 1, 1, 2, 1, 2, 1, 2, 2, 1, 1, 3, 1, 1, 2, 1, 1, 2, 2, 1, 2, 1, 2, 2, 1, 1, 3, 1, 1, 1, 1, 1, 1, 4, 1, 1, 2, 3, 1, 1, 3, 2, 1, 1, 4, 1, 1, 2, 1, 3, 1, 2, 2, 2, 1, 2, 3, 1, 1, 3, 1, 2, 1, 3, 2, 1, 1, 4, 1, 1, 2, 1, 1, 3, 2, 1, 2, 2, 2, 1, 3, 1, 2, 2, 1, 2, 2, 2, 2, 1, 2, 3, 1, 1, 3, 1, 1, 2, 3, 1, 2, 1, 3, 2, 1, 1, 4, 1, 1, 1

Offset: 1

Clark Kimberling, Sep 07 2000

nonn

			Flatten the list of ordered lattice points, (1,1,1,1) < (1,1,1,2) < (1,1,2,1) < ... as 1,1,1,1, 1,1,1,2, 1,1,2,1, ...

Cf. A057554, A057555, A057556, A057557, A057558, A186006.

lexicographicLattice[{dim_,maxHeight_}]:= Flatten[Array[Sort@Flatten[(Permutations[#1]&)/@IntegerPartitions[#1+dim-1,{dim}],1]&,maxHeight],1]; Flatten@lexicographicLattice[{4,4}]
(* by Peter J. C. Moses, Feb 10 2011 *)

Extended by Clark Kimberling, Feb 10 2011

A057554 Lexicographic ordering of MxM, where M={0,1,2,...}.

Original entry on oeis.org

0, 0, 0, 1, 1, 0, 0, 2, 1, 1, 2, 0, 0, 3, 1, 2, 2, 1, 3, 0, 0, 4, 1, 3, 2, 2, 3, 1, 4, 0, 0, 5, 1, 4, 2, 3, 3, 2, 4, 1, 5, 0, 0, 6, 1, 5, 2, 4, 3, 3, 4, 2, 5, 1, 6, 0, 0, 7, 1, 6, 2, 5, 3, 4, 4, 3, 5, 2, 6, 1, 7, 0, 0, 8, 1, 7, 2, 6, 3, 5, 4, 4, 5, 3, 6, 2, 7, 1, 8, 0, 0, 9, 1, 8, 2, 7, 3, 6, 4, 5, 5, 4, 6, 3, 7, 2, 8, 1, 9, 0, 0, 10, 1, 9, 2, 8, 3, 7, 4, 6, 5, 5, 6, 4, 7, 3, 8, 2, 9, 1, 10, 0, 0, 11, 1, 10, 2, 9, 3, 8, 4, 7, 5, 6, 6, 5, 7, 4, 8, 3, 9, 2, 10, 1, 11, 0

Offset: 1

Clark Kimberling, Sep 07 2000

nonn

A057555 gives the lexicographic ordering of N x N, where N={1,2,3,...}.

			Flatten the ordered lattice points: (0,0) < (0,1) < (1,0) < (0,2) < (1,1) < ... as 0,0, 0,1, 1,0, 0,2, 1,1, ...

Alois P. Heinz, Table of n, a(n) for n = 1..20022

Cf. A057555, A057556, A057557, A057558, A057559.

lexicographicLattice[{dim_,maxHeight_}]:= Flatten[Array[Sort@Flatten[(Permutations[#1]&)/@IntegerPartitions[#1+dim-1,{dim}],1]&,maxHeight],1]; Flatten@lexicographicLattice[{2,12}]-1 (* Peter J. C. Moses, Feb 10 2011 *)

[l for i in range(20) for k in range(i,-1,-1) for l in (i-k, k)] # Nicholas Stefan Georgescu, Oct 10 2023

A057558 Lexicographic ordering of MxMxMxM, where M={0,1,2,...}.

Original entry on oeis.org

0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 1, 0, 0, 1, 0, 0, 1, 0, 0, 0, 0, 0, 0, 2, 0, 0, 1, 1, 0, 0, 2, 0, 0, 1, 0, 1, 0, 1, 1, 0, 0, 2, 0, 0, 1, 0, 0, 1, 1, 0, 1, 0, 1, 1, 0, 0, 2, 0, 0, 0, 0, 0, 0, 3, 0, 0, 1, 2, 0, 0, 2, 1, 0, 0, 3, 0, 0, 1, 0, 2, 0, 1, 1, 1, 0, 1, 2, 0, 0, 2, 0, 1, 0, 2, 1, 0, 0, 3, 0, 0, 1, 0, 0, 2, 1, 0, 1, 1, 1, 0, 2, 0, 1, 1, 0, 1, 1, 1, 1, 0, 1, 2, 0, 0, 2, 0, 0, 1, 2, 0, 1, 0, 2, 1, 0, 0, 3, 0, 0, 0

Offset: 1

Clark Kimberling, Sep 07 2000

nonn

			Flatten the list of ordered lattice points, (0,0,0,0) < (0,0,0,1) < (0,0,1,0) < ... as 0,0,0,0, 0,0,0,1, 0,0,1,0, ...

Alois P. Heinz, Table of n, a(n) for n = 1..19380

Cf. A057556, A057557, A057559.

lexicographicLattice[{dim_,maxHeight_}]:= Flatten[Array[Sort@Flatten[(Permutations[#1]&)/@IntegerPartitions[#1+dim-1,{dim}],1]&,maxHeight],1]; Flatten@lexicographicLattice[{4,4}]-1
(* by Peter J. C. Moses, Feb 10 2011 *)

Extended by Clark Kimberling, Feb 10 2011

A186006 Lexicographic ordering of N x N x N x N x N, where N={1,2,3,...}.

Original entry on oeis.org

1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 2, 1, 1, 1, 2, 1, 1, 1, 2, 1, 1, 1, 2, 1, 1, 1, 1, 1, 1, 1, 1, 3, 1, 1, 1, 2, 2, 1, 1, 1, 3, 1, 1, 1, 2, 1, 2, 1, 1, 2, 2, 1, 1, 1, 3, 1, 1, 1, 2, 1, 1, 2, 1, 2, 1, 2, 1, 1, 2, 2, 1, 1, 1, 3, 1, 1, 1, 2, 1, 1, 1, 2, 2, 1, 1, 2, 1, 2, 1, 2, 1, 1, 2, 2, 1, 1, 1, 3, 1, 1, 1, 1, 1, 1, 1, 1, 4, 1, 1, 1, 2, 3, 1, 1, 1, 3, 2, 1, 1, 1, 4, 1, 1, 1, 2, 1, 3, 1, 1, 2, 2, 2, 1, 1, 2, 3, 1, 1, 1, 3, 1, 2, 1, 1, 3, 2, 1, 1, 1, 4, 1, 1, 1, 2, 1, 1, 3

Offset: 1

Clark Kimberling, Feb 10 2011

nonn

By changing a single number, the Mathematica code suffices for other dimensions: N x N (A057555), N x N x N (A057557), N x N x N x N (A057559), and higher.

			First, list the 5-tuples in lexicographic order: (1,1,1,1,1) < (1,1,1,1,2) < (1,1,1,2,1) < (1,1,2,1,1) < ... < (1,2,2,1,1) < (1,1,3,1,1) < ... Then flatten the list, leaving 1,1,1,1,1, 1,1,1,1,2, 1,1,1,2,1, 1,1,2,1,1, ...

G. C. Greubel, Table of n, a(n) for n = 1..1000

Cf. A057555, A057557, A057559.

lexicographicLattice[{dim_,maxHeight_}]:= Flatten[Array[Sort@Flatten[(Permutations[#1]&)/@IntegerPartitions[#1+dim-1,{dim}],1]&,maxHeight],1];
lexicographicLatticeHeightArray[{dim_,maxHeight_,axis_}]:= Array[Flatten@Position[Map[#[[axis]]&, lexicographicLattice[{dim,maxHeight}]],#]&,maxHeight]
Take[Flatten@lexicographicLattice[{5,12}],160]
(* Peter J. C. Moses, Feb 10 2011 *)

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A186003 Distance array associated with ordering A057557 of N X N X N, by antidiagonals (distances to yz plane).

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A186004 Distance array associated with ordering A057557 of N X N X N, by antidiagonals (distances to xz plane).

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A186005 Distance array associated with ordering A057557 of N X N X N by antidiagonals (distances to xy plane).

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A057555 Lexicographic ordering of N x N, where N = {1,2,3...}.

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A057556 Lexicographic ordering of M x M x M, where M={0,1,2,...}.

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A057559 Lexicographic ordering of NxNxNxN, where N={1,2,3,...}.

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A057554 Lexicographic ordering of MxM, where M={0,1,2,...}.

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A057558 Lexicographic ordering of MxMxMxM, where M={0,1,2,...}.

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